[Theorem 3.1 of [1]}, modified]
Let \(f\)  satisfy Assumption , and let \((x)\)  be the Hessian at \(x\)  having range that spans whole \(^d\)  for all \(x\) . Then
\( _{\hat{S}}[ f(x_{k+1}) - f(x^*) ] \le \left( 1 - \frac{\hat{\sigma }(\hat{S},x_k)\mu }{L} \right) (f(x_k) - f(x^*), \) 
